Physics Motivation

The final choice will also be influenced by operating costs. A normal coil will perforce be in the 5-6 MW range a superconducting coil consumes far less power, but incurs ongoing operating costs associated with its cryogenics. 

An instrumented flux return can serve as a calorimeter for detection and to extend effective tt/// separation to lower than usual momenta. In this regard, the larger thickness of a normal coil in interaction lengths becomes a consideration. 

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In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

The corresponding gradient-corrected correlation functionals have even more complicated analytical forms and cannot be understood by simple physically motivated reasonings. We therefore refrain from giving their explicit expressions and limit ourselves to a more qualitative discussion of the most popular functionals. Among the most widely used choices is the correlation counterpart of the 1986 Perdew exchange functional, usually termed P or P86. This functional employs an empirical parameter, which was fitted to the... 

A useful tool for the development of physically motivated approximate models of solution thermodynamics, particularly in view of quasichemical extensions... 

Similar values for the maximal mass and radius were found in a perturbative approach with a physically motivated choice of the renormalization scale [17]. In a Schwinger-Dyson approach [18], M rb 0.7Msun and R 9 km were obtained. 

The aim of this paper is ascertain whether it is possible to determine the ground state second-order correlation energy of the hydrogen molecule to sub-millihartree accuracy using a basis set containing only s-type Gaussian functions with exponents and distribution determined by an empirical, but physically motivated, procedure. 

It has been shown that the second order electron correlation energy for the ground state of the hydrogen molecule at its equilibriiun nuclear geometry can be described to an accmacy below the sub-milliHartree level using a distributed basis set of Gaussian basis subsets containing only s-type functions only. Each of the basis subsets are taken to be even-tempered sets. The distribution of the subsets is empirical but nevertheless physically motivated. 

In this paper we review some of our recent work on the dynamics of step bunching and faceting on vicinal surfaces below the roughening temperature, concentrating on several cases where interesting two dimensional (2D) step patterns form as a result of kinetic processes. We show that they can be understood from a unified point of view based on an approximate but physically motivated extension to 2D of the kind of ID step models studied by a number of workers. For some early examples, see refs. [1-5]. We have tried to make the conceptual and physical foundations of our own approach clear, but have made no attempt to provide a comprehensive review of work in this active area. More general discussions from a similar perspective and a guide to the literature can be found in recent reviews by Williams and Williams and BartelF. 

The physical motivation behind this choice is that S now becomes an antisymmetric observable under time reversal. Albeit 5 p(r) is always antisymmetric, the choice of Eq. (25) is the only one that guarantees that the total dissipation S changes sign upon reversal of the path, iS(T ) = —iS(T). The symmetry property of observables under time reversal and the possibility of considering boundary terms where S is symmetric (rather than antisymmetric) under time reversal has been discussed in Ref. 43. 

Anomalous rectification [3]. Our aim in this section is to show that under certain conditions development of a nonequilibrium space charge may yield, besides the punch through, some additional effects, unpredictable by the locally electro-neutral formulations. We shall exemplify this by considering two parallel formulations—the full space charge one and its locally electro-neutral counterpart. It will be observed that inclusion of the space charge into consideration enables us to account for the anomalous rectification effect that could not be predicted by the locally electro-neutral treatment. Physical motivation for this study is as follows. 

Because we do not now have available eigenfunctions of the Hamiltonian of any large molecule, it is necessary to proceed using physically motivated approximations. It is then also convenient to require that ... 

If the TV-particle basis were a complete set of JV-electron functions, the use of the variational approach would introduce no error, because the true wave function could be expanded exactly in such a basis. However, such a basis would be of infinite dimension, creating practical difficulties. In practice, therefore, we must work with incomplete IV-particle basis sets. This is one of our major practical approximations. In addition, we have not addressed the question of how to construct the W-particle basis. There are no doubt many physically motivated possibilities, including functions that explicitly involve the interelectronic coordinates. However, any useful choice of function must allow for practical evaluation of the JV-electron integrals of Eq. 1.7 (and Eq. 1.8 if the functions are nonorthogonal). This rules out many of the physically motivated choices that are known, as well as many other possibilities. Almost universally, the iV-particle basis functions are taken as linear combinations of products of one-electron functions — orbitals. Such linear combinations are usually antisymmetrized to account for the permutational symmetry of the wave function, and may be spin- and symmetry-adapted, as discussed elsewhere ... 

The equations for the two-phase fully mixed system are thus reduced to the equations for a single stirred tank by the physically motivated notion of only using the available fraction of the feed. This has been made possible by the uniformity of the dense phase and the linearity of the transfer process in the bubble. This allows us to see how the rather implausible assumption that the bubble phase is really well mixed can be made more realistic. Let us go to the other extreme, and suppose that the bubbles ascend with uniform velocity U. The surface area per unit length of reactor is SIH, where 5 is, as before, the total interphase area and H the height of the bed. If h is the transfer coefficient and z the height of a given point, a balance over the interval (z, z + dz) gives the equation for the concentration in the bubble phase b(z)... 

Configurational-integral ratios of the form of Eq. (50) appear widely in the free-energy literature, but the underlying physical motivation is not always the same. The spectrum of possible usages is covered by writing Eq. (50) in the more general form... 

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